We define tree-cycle graphs (see Definition REF ) as graphs built out of trees through one of two operations. Through a judicious use of the colored \(\beta \)  polynomial, we may reduce a large class of planar graphs to tree-cycle graphs without changing \(\beta (-1)\) . We view (d), which can be generalized to certain non-planar graphs, as the emergence of the Euler characteristic (which is a global topological invariant). This has been shown by some of us recently [1]} via computations of multipartite information on a plane, which capture the Topological Entanglement Entropy (TEE) [2]}, [3]} of a topologically ordered phases of quantum matter [4]}, [5]}, [6]}. We also note that a recent work [7]} has shown that the multipartite information between \(D+1\)  partitions of a \(D\) -dimensional non-interacting Fermi gas is proportional to the Euler characteristic of the \(D\) -dimensional Fermi volume.
